Sin(45°)

Definition

sin(45°) is the sine of a 45-degree angle, which is equal to $$\frac{\sqrt{2}}{2}$$. This value represents the ratio of the length of the opposite side to the hypotenuse in a right triangle, specifically in the context of a 45-45-90 triangle where both legs are of equal length. The connection to the unit circle allows us to visualize this angle and its sine value as a coordinate point on the circle.

5 Must Know Facts For Your Next Test

1. In a 45-45-90 triangle, both legs are equal in length, allowing sin(45°) to be calculated as $$\frac{\text{leg}}{\text{hypotenuse}}$$, leading to the value $$\frac{\sqrt{2}}{2}$$.
2. On the unit circle, sin(45°) corresponds to the y-coordinate of the point where the terminal side of the angle intersects the circle.
3. The value of sin(45°) is approximately 0.7071, which is often used in applications involving angles and distances.
4. Sine is an odd function, meaning that sin(-x) = -sin(x), so sin(-45°) will equal -sin(45°).
5. The sine function is periodic with a period of 360°, meaning that sin(405°) also equals sin(45°).

Review Questions

• How does sin(45°) relate to the characteristics of a 45-45-90 triangle?
  • In a 45-45-90 triangle, both legs are equal, and using the Pythagorean theorem, we can derive that each leg has a length of 'x' while the hypotenuse is 'x√2'. The sine function gives us sin(45°) = opposite/hypotenuse, which translates to $$\frac{x}{x√2} = \frac{1}{\sqrt{2}}$$. This simplifies to $$\frac{\sqrt{2}}{2}$$ when rationalized. Thus, understanding sin(45°) helps us comprehend how angles relate to specific side lengths in special triangles.
• Describe how the value of sin(45°) can be visualized using the unit circle.
  • On the unit circle, an angle of 45° corresponds to a point located at $$\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$$. The x-coordinate represents cos(45°), while the y-coordinate represents sin(45°). Since the radius of the unit circle is 1, this visual representation confirms that sin(45°) equals $$\frac{\sqrt{2}}{2}$$ because it is directly related to the vertical distance from the x-axis at this angle. This understanding links geometric concepts with trigonometric functions.
• Evaluate how knowing sin(45°) can be applied in real-world situations or advanced mathematics.
  • Knowing that sin(45°) = $$\frac{\sqrt{2}}{2}$$ has practical implications in fields such as engineering and physics, especially when dealing with diagonal forces or components in vector analysis. For example, if you have a force acting at a 45-degree angle on an object, you can use this sine value to determine both horizontal and vertical components effectively. Furthermore, it also aids in solving problems involving periodic functions and oscillations where angles frequently appear.

Related terms

Unit Circle: A circle with a radius of one, centered at the origin of a coordinate plane, used to define trigonometric functions for all angles.

45-45-90 Triangle: A special right triangle where the two legs are congruent and the angles are 45 degrees each, resulting in specific side length ratios.

Trigonometric Ratios: The ratios of the lengths of the sides of a right triangle, which include sine, cosine, and tangent, and are used to relate angles to side lengths.